In Journey to Babel, one of the dramatic points of the episode is that Sarek needs an operation to be conducted and Dr McCoy performs this, despite not having any actual surgical experience with a Vulcan. I've got a copy of Star Trek Chronology which shows that in the next episode, Dr M'Benga takes care of Spock, specifically because he interned in a Vulcan ward. As in my copy of Chronology it is not established when M'Benga was assigned to the Enterprise and there is no indication that the Enterprise took on any crew between Journey to Babel and A Private Little War, I'm assuming that M'Benga was already on the Enterprise. So, why would they risk using McCoy (no disrespect to the man) when they had the option of using Dr M'Benga who probably had more experience than McCoy, unless my assumption about M'Benga already being assigned is incorrect? I'm looking for the in-universe reason; I presume the out-of-universe reason is that M'Benga's character hadn't been developed yet.
The Memory Alpha page doesn't provide much illumination on this matter, providing basically the same as Star Trek Chronology. The Memory Beta page, however, helps fill in some of the gaps. As there is no canon information to explain this, I will use the contents from Memory Beta instead.
Some background first: The events of Journey to Babel and A Private Little War both occurred in 2267.
The Memory Beta page is a little vague on details, but it would seem that M'Benga was assigned to the Enterprise sometime prior to 2267 in a temporary capacity. In 2267, however, M'Benga returned to Vulcan to further his studies. M'Benga then returned to the Enterprise whilst they were on a mission to develop a cure for a deadly plague on Nisus. It was only following that event that M'Benga was offered a permanent placement on the Enterprise by McCoy. Then:
a few weeks after his re-assignment to the Enterprise, Spock was badly injured...
i.e. the events of A Private Little War.
So, in summary, it would seem that M'Benga was on the Enterprise sometime before Journey to Babel, but left for Vulcan prior to the events of that episode, explaining why he was not chosen to complete the operation on Sarek during that episode.